Toads and Tessellations

$Toads and Tessellations book cover$

Grade 4
5 skill lessons

Math

Skill lessons

The mathematics the book carries, taught directly. Printable workbook pages and teacher keys are not available for this book yet.

1The Angles That Fill a Point
2No Gap or Overlap
3Flip, Turn, Slide
4The Angle That Closes the Gap
5Twelve Pairs from One Piece

Math + reading

Math through the reading

The same book, entered through a sentence it prints. Change one word in the reading and the numbers change with it.

No math-through-reading lessons for this book yet.

Reading

Reading and language lessons

The same book read as a text. What a word means in its sentence, how a character is shown, how the writing is put together - no mathematics.

No reading and language lessons for this book yet.

Both exports carry any edits you have made. Edits are saved in this browser only — nobody else sees them, and they are not sent anywhere.

Skill lessons

5 lessons on the mathematics the book carries. Printable workbook pages and teacher keys are not available for this book yet.

1. The Angles That Fill a Point Concept · Measuring the angles that meet at one point and testing whether they sum to 360 degrees

Aida states the rule in one sentence and the book then has the characters use it: the internal angles of every shape meeting at a point add up to 360 degrees. Children lay pattern pieces round a marked dot, measure each angle that touches the dot, and add their measures. The idea under it: a child finds out that a point either closes or does not, and that the sum is what decides it, rather than how the arrangement looks.

Full lesson — 20 minutes

1. Measure one piece's corners (5 min)
Take one paper triangle and one paper rectangle and measure every corner with the angle scale, writing each measure on the piece. Check your rectangle's four corners against each other before you go on.
Word preview: internal angle, degrees
The internal angles of all the shapes at that point add up to 360 degrees and form a perigon, or round angle.
2. Build a point and add the angles (6 min)
Mark a dot on a sheet. Lay pieces round the dot so their corners meet at it and none overlaps. Measure every angle that touches the dot and add the measures. Say what your total came to.
Word preview: fill a point
We can flip, turn, or rotate pieces any way we want in order to completely fill a point.
3. Name what you built (5 min)
Read what Aida calls a closed point and write that name beside your own total. Then say in one sentence whether your arrangement earned the name, and how your total says so.
Word preview: perigon
A perigon is a 360-degree angle,
4. Find the point that does not close (4 min)
Take the arrangement on the card that has been built for you, measure the angles at its dot and add them. Say how far the total is from a closed point and put a finger on where the gap is.
Word preview: gap

Short version — 9 minutes

1. Build a point and add the angles (5 min)
Lay pieces round a marked dot so their corners meet with no overlap, measure every angle at the dot and add the measures.
We can flip, turn, or rotate pieces any way we want in order to completely fill a point.
2. Find the point that does not close (4 min)
Measure and add the angles at the dot on the prepared card, and say how far the total is from a closed point.

Three levels

Support: Use the three pieces whose angles are 90, 90 and 180 so the first total closes cleanly, and read the scale together for the first measure. The child still measures every angle and still adds them.
At Grade: Two pieces measured all round, a point built and its angles summed, the name written beside the total, and the prepared 340-degree card measured and its gap located.
Extension: Build a point three different ways using a different number of pieces each time, and say what stayed the same about the three totals when everything else changed.

English learners

Measuring transfers; a child who can read the scale can read it in any language. What will not carry over is that internal names the angle inside the shape at that corner, not the shape's inside generally — a distinction English makes with one adjective. Point at the corner, then at the whole piece, and say internal angle only while pointing at the corner. Rehearsable frame: The angles at this point add up to ____, so the point ____. Children may add in their strongest language and then say the frame in English with the arrangement in front of them.

Materials

A set of paper pattern pieces per pair: four triangles, three rectangles and two trapezoids, all with straight sides
An angle scale marked in whole degrees, per child
A pencil
A plain sheet with one marked dot, per pair
The two sentences about the angles at a point, enlarged
A prepared arrangement card per pair whose angles at the dot sum to 340 degrees
Word cards: internal angle, degrees, perigon, point

Standards & curriculum

These lessons are being matched against curricula now.

2. No Gap or Overlap Application · Testing whether a repeated shape covers a surface with no gap and no overlap

Tessel's objection is the book's real problem: shoe pieces have curves and leave scraps. Aida's answer is to trim them to straight-sided shapes, and Enzo's is that the result repeats with no gap or overlap. Children trim, tile a frame, and count squares covered and squares left. The covering test gets run on a real sheet here, where the answer is a count and not an impression.

Full lesson — 20 minutes

1. Trim the pieces (5 min)
Take the curved shoe-piece outlines and trim each one into straight-sided rectangles and triangles, keeping every scrap you cut off in the scrap tray.
Word preview: trim
She trimmed the shoe pieces into rectangles and triangles.
2. Tile the frame (6 min)
Fill the frame with your trimmed pieces, flipping, turning and sliding them as you need to. Every time two edges meet, check that no light shows between them and that no piece lies on top of another.
Word preview: gap, overlap
No gap or overlap—a new form in a snap!
3. Count what covered and what was left (5 min)
Count the squares your pieces covered and the squares still showing, and write both counts. Then count the scraps in your tray and say whether Tessel was right about this arrangement.
There are always leftover scraps.
4. Repeat the unit across the sheet (4 min)
Take the group of pieces that filled one part of your frame and repeat it along the sheet as far as the paper allows. Say how many times it repeated and whether the pattern could keep going.
The pattern could go on forever.

Short version — 9 minutes

1. Tile the frame (5 min)
Fill the squared frame with the trimmed pieces, checking at every meeting of edges that no light shows and no piece overlaps another.
No gap or overlap—a new form in a snap!
2. Count what covered and what was left (4 min)
Count and write the squares covered and the squares still showing, and say whether any scrap was left over.
There are always leftover scraps.

Three levels

Support: Supply the pieces already trimmed and use a frame six squares by five. The child still tiles, still checks every meeting of edges, and still counts both boxes.
At Grade: Four outlines trimmed, the twelve-by-ten frame tiled, both counts written, the scraps counted, and the unit repeated along a strip with the number of repeats recorded.
Extension: Change one piece's shape so the frame can no longer be filled, then say which edge stopped fitting and what that does to the count of squares showing.

English learners

Tiling and counting transfer. Gap and overlap do not, and they are opposites a new listener will not hear as a pair — one is an absence and one is a doubling. Teach both with the frame: slide two pieces apart and say gap, then lay one on the other and say overlap. Rehearsable frame: There is no ____ here because the edges ____. Children may report the fit in their strongest language and then say the frame in English with a finger on the join.

Materials

Four curved shoe-piece outlines per pair, printed on squared paper
Scissors per pair
A scrap tray per pair
A squared frame per pair, twelve squares by ten
A two-box recording card per pair, headed covered and showing
A long strip of squared paper per pair
Word cards: trim, gap, overlap, tile

Standards & curriculum

These lessons are being matched against curricula now.

3. Flip, Turn, Slide Representation · Naming and recording which of flip, turn and slide placed a piece

The book names the moves twice over. Enzo may flip, turn or rotate any piece to fill a point; later, when the toad has bitten a rectangle, he slides the bitten chunk across without rotating or flipping it. Children perform each move on one piece, record which move produced each placement, and then have to undo a partner's move. The representation is the record: a placement with the move that made it written beside it.

Full lesson — 20 minutes

1. Three moves, three cards (5 min)
Put the flip, turn and slide cards in front of you. Take one L-shaped piece and perform each named move on it in turn, leaving the piece where the move put it before you pick the next card.
Word preview: flip, turn, slide
We can flip, turn, or rotate pieces any way we want in order to completely fill a point.
2. The slide that changes nothing else (6 min)
Cut a notch from one side of a paper rectangle and slide the cut chunk to the opposite side without turning it or flipping it over. Check that the dot is still facing you, then trace the new outline on the grid.
I can slide the nibbled piece to the other side of the rectangle without rotating or flipping it.
3. Record the move for each placement (5 min)
Make four placements of the L-shaped piece on the grid, tracing each one and writing beside it which single move got the piece there from the placement before. One of your four must use the slide.
4. Undo a partner's move (4 min)
Swap grids with another pair. For one of their traced placements, work out the move that would put the piece back where it came from, and say whether that move has the same name as theirs.

Short version — 9 minutes

1. The slide that changes nothing else (5 min)
Cut a notch from one side of a rectangle and slide the chunk to the opposite side without turning or flipping it, then trace the new outline.
I can slide the nibbled piece to the other side of the rectangle without rotating or flipping it.
2. Record the move for each placement (4 min)
Trace three placements of the piece and write beside each one which single move got it there from the placement before.

Three levels

Support: Work with the rectangle and the slide only, and mark the front of the piece with a large dot so a flip is unmistakable. The child still names the move and still traces the result.
At Grade: All three moves performed and named, the notched rectangle slid and traced, four placements recorded with their moves, and one of a partner's moves undone.
Extension: Find a placement that two different single moves could both have produced, and say what is true about the piece that lets one placement have two names.

English learners

Doing the move transfers immediately, and the three words are picturable if they are taught on the piece rather than on paper. The part that has to be taught is that English uses turn both for the action and for the whole rotation, so a child hears one word for two things. Use the move cards every time and touch the card while doing the move. Rehearsable frame: I ____ the piece, so it is now ____. Children may narrate the move in their strongest language first, then say the frame in English while the piece is still in the new place.

Materials

Three move cards per pair, marked flip, turn and slide
One L-shaped paper piece per pair, marked with a dot on the front
A grid sheet per pair
A paper rectangle and scissors per pair
A pencil
Word cards: flip, turn, slide, rotate

Standards & curriculum

These lessons are being matched against curricula now.

4. The Angle That Closes the Gap Fluency · Subtracting a sum of angles from 360 to find the angle that closes a point

Once children accept that a closed point measures 360 degrees, the arithmetic follows: add what is there, subtract from 360, and the difference is the piece still needed. Children work five prepared points, find each missing angle, and then cut a piece to that measure and check it closes. Practised until the subtraction is quick and the check never varies: lay the cut piece in and look for light.

Full lesson — 20 minutes

1. Add what is already there (5 min)
For each of the five prepared points, measure the angles at the dot and write their total in the first column. Do all five totals before you subtract anything.
The internal angles of all the shapes at that point add up to 360 degrees and form a perigon, or round angle.
2. Subtract from three hundred sixty (6 min)
Write 360 at the head of the second column and subtract each total from it. Put the difference in the third column. Say what the number in the third column is the measure of.
Word preview: difference
3. Cut the piece that fits (5 min)
Choose one of your five points and cut a paper wedge whose angle measures the difference you found. Measure your wedge before you place it.
4. Check by closing the point (4 min)
Lay your wedge into the gap and hold the card up to the light. If light shows, measure the wedge and the gap again and say which of the two was wrong.
We can flip, turn, or rotate pieces any way we want in order to completely fill a point.

Short version — 9 minutes

1. Add what is already there (5 min)
Measure and total the angles at the dot for three of the prepared point cards, writing each total in the first column.
The internal angles of all the shapes at that point add up to 360 degrees and form a perigon, or round angle.
2. Subtract from three hundred sixty (4 min)
Subtract each total from 360 and say what the difference is the measure of.

Three levels

Support: Work the two cards whose gaps are 90 and 120, and supply the measured total so the child's work starts at the subtraction. Cutting and checking the wedge stay.
At Grade: Five points measured, totalled and subtracted from 360, one wedge cut to the difference, measured, and checked against the light.
Extension: Fill one gap with two wedges instead of one, choosing measures that add to the difference, and say how many other pairs of wedges would have worked.

English learners

Subtraction transfers. The English that does not is the word difference doing double duty — in ordinary speech it means how things are unalike, and here it names a measure a child is about to cut out. Say it only while pointing at the gap, and put the word on a card beside the third column. Rehearsable frame: 360 take away ____ leaves ____, so the gap is ____ degrees. Children may compute in their strongest language and then say the frame in English with their recording sheet in view.

Materials

Five prepared point cards per pair, with gaps of 30, 45, 60, 90 and 120 degrees
An angle scale marked in whole degrees, per child
A five-row recording sheet ruled for three columns, per pair
A pencil
Paper and scissors per pair
A window or lamp to hold cards against
Word cards: difference, degrees, gap

Standards & curriculum

These lessons are being matched against curricula now.

5. Twelve Pairs from One Piece Reasoning · Multiplying pairs to items and deciding whether one supply covers the whole order

The order in this book is stated in pairs and the leather is stated as one piece, and the two never meet in a number anywhere. Children turn the pairs into shoes, count the pieces one shoe takes, multiply up to the whole order, and then decide from their own tiling whether one sheet could hold it. The demand is the decision at the end, which has to be funded by the child's own multiplication rather than by what the story reports.

Full lesson — 20 minutes

1. How many shoes is twelve pairs (5 min)
Read what the customer has ordered and work out how many single shoes that is. Write the multiplication you used, not just the answer.
Word preview: pair
She wants twelve pairs of shoes made out of one piece of leather.
all twelve princesses must have a pair, or I will not pay!
2. Count the pieces one shoe needs (5 min)
Take one trimmed shoe outline and count the straight-sided pieces it was cut into. Write that count on the second line of your sheet.
She trimmed the shoe pieces into rectangles and triangles.
3. Multiply up to the whole order (6 min)
Use your two numbers to find how many pieces the whole order needs. Write the multiplication on the third line and check it by doubling the count for one pair twelve times or by a method you can explain.
4. Decide whether the leather can do it (4 min)
Look at how many pieces your own tiled frame held. Take a side on the question the book asks and say it out loud, with your number from the third line inside the sentence.
can we make twelve pairs of shoes using only this piece of leather?

Short version — 9 minutes

1. How many shoes is twelve pairs (5 min)
Read the order and work out how many single shoes it is, writing the multiplication used and not only the answer.
She wants twelve pairs of shoes made out of one piece of leather.
2. Decide whether the leather can do it (4 min)
Take a side on the book's question and say it aloud with your own number inside the sentence.
can we make twelve pairs of shoes using only this piece of leather?

Three levels

Support: Lay out twelve paper pairs so the child can count the shoes before multiplying, and keep the pieces-per-shoe count to a shoe cut into three. Writing the multiplication and taking a side stay.
At Grade: Both numbers found and written, the whole-order product computed and checked a second way, and a side taken aloud with the product in the sentence.
Extension: Work out how many pieces would be needed if each shoe were cut into one more piece, and say how much the whole order changes for one extra cut per shoe.

English learners

Multiplying transfers. The English that does not is pair, which names two things as one unit and is easily heard as a single object — a child who reads twelve pairs as twelve shoes gets a defensible-looking wrong answer. Put two paper shoes under one card marked pair and keep it in view. Rehearsable frame: Twelve pairs is ____ shoes, because one pair is ____. Children may work the count in their strongest language and then say the frame in English.

Materials

The two ordering sentences, enlarged
A recording sheet ruled for three lines, per child
A pencil
One trimmed shoe outline per child
The book's question, enlarged
Word cards: pair, order, piece

Standards & curriculum

These lessons are being matched against curricula now.